In the nineteenth century prodigious skills of arithmetic computation was a pre requisite for top employment.
During the course of the twentieth century the need for this skill diminished as technology was able to automate arithmetic. In the 21st Century we still insist that the calculation skill is somehow the basis of the rest of mathematics. If a child cannot learn arithmetic, an obsolete skill, he will have a major problem in school. Suddenly poor computational skills becomes "math disability".
I do not advocate that schools abolish teaching computation. Let there be arithmetic courses, but also let there be mathematical courses where arithmetic is invisible. How you make arithmetic invisible is a matter of taste, but let me share mine.
Use a spreadsheet to present problems in arithmetic facts like an worksheet but with instant feedback and automatic storage of the results. Use a spreadsheet to teach and sharpen the estimation skills of the student.
At the same time you show the commands and formulas you can write in a spreadsheet to automatically get the answers in the worksheet. That child has learned the arithmetic facts and the spreadsheet is an electronic manipulative.
That is the way I would teach the Theory and Practice of Arithmetic
During the rest of Mathematics, arithmetic is invisible and the student is given work problems and uses the spreadsheet to solve them.
For example,
Teach them to use Goal Seek to solve one variable algebra and solver for mutlivariable. If they know Algebra enough to write those equations in a spreadsheet then let the spreadsheet take care of the arithmetic.
Another example,
A spreadsheet can produce 2 and 3 dimensional objects that could be used in the more graphic mathematics as Geometry and Trigonometry. The cosine and sine functions are arithmetic free but you have to know when to use them.
This is, of course, is merely a preliminary outline, but the thrust will not change. Teach arithmetic as a desirable skill but teach other parts of mathematics where the student can take the arithmetic for granted. Just like in the workplace of the 21st Century.
Adding Spreadsheets To Early Mathematics / Arithmetic
By sherose (not verified)I am comfident that the time has come to add the spreadsheet to the early development of mathematical skills. Note the deeper comprehension of place value concepts when adding ten to a range of numbers i.e.: 10, 100, 1000, 10,000, 100,000.
Teachers slave over place value development; it is a deeper understanding of mathematics for which the spreadsheet is eminently designed. Currently, teachers utilize large charts (similar to a spreadsheet). The comprehension of numbers below the decimal can also be refined by having students estimate and then view on a spreadsheet. In classes where I do not have access to a computer, a set of calculators is used for this process.
In fact, NY State uses a calculator for mathematics assessment; however, students are assessed for their mastery following comprehension. Graphing calculators are utilized currently, at the eighth grade level. Parents are responsible for their purchase.
Young children can predict the third angle of a triangle given
the numbers/size of the other two. This is one of the many steps of early strategies that are needed to develop mathematical concepts. Hands-on is clearly first in any discovery process, but the reverse (spreadsheet before manipulatives) will be needed, as well. I recall my own three year old daughter pulling out shoes to demonstrate the comprehension of forming a square with the early learning MIT computer program on an ATARI (abstract to concrete).
The spreadsheets are for students in kindergarten through the middle gradesin basic concept development. Note the current use of geoboards (a pegboard with rubberbands) to predict and to demonstrate concepts i.e. 3 peg triangle compared to a 9 peg triangle or 2 x 2 square compared to a 3 x3 square.
Clearly, some students need to work for longer periods at concrete levels using cuisenaire rods or stern materials to develop concepts that others grasp immediately. We have even used cardboard boxes on for carpentry to develop basic mathematical content. Surely, the spreadsheet should be introduced even if the child is only ready for adding one (+1) to each digit in the range. I recall the third graders seated beside their parents on Family Math Day in the 90's; they were responding effectively as Dr. Z encouraged them to work with six digit number. I noted that the parents were delighted with their offspring's enthusiasm, concentration and growth; none of the parents assumed that the multiplication tables were mastered in the process.
De-Emphasizing Computational Skills - Yes or No
By csonitaI also used to believe that mathematical processes were more important than computational skills. Then I started working in the field of Adult Education - an area in which the teacher is assessed on his ability to help his students obtain gainful employment.
Although I still believe teachers must become more effective at helping students understand and master mathematics concepts and procedures, a significant number of employers want employees who can calculate in their heads. I'm not talking about cashiers. I'm talking about carpenters, electricians, masons, auto mechanics, plumbers, etc. Educators have got to find the time to help students master the abstract and use the concrete.
De-emphasizing arithmetic is not ignoring arithmetic
By drzIn the original message I wrote:
I do not advocate that schools abolish teaching computation. Let there be arithmetic courses, but also let there be mathematical courses where arithmetic is invisible.
I would like a bit of clarification here. csonita wrote:
a significant number of employers want employees who can calculate in their heads. I'm not talking about cashiers. I'm talking about carpenters, electricians, masons, auto mechanics, plumbers, etc.
What precision of calculation are we talking about? Are carpenters expected to know how many 2x4 pieces of lumber to purchase or exactly how many running inches to 3 decimal places the job will take? It is is the former, then we need to strengthen estimation skills. The latter demands the use of a calculator unless time is not important.
In Singapore and Finland....
By JustinecDr. Z,
In Singapore and Finland, two of the main countries leading the world overall in mathematics, they teach the concepts in groups in word problem format in the classroom. The teachers are facilitators, not teachers for the whole class time like here in the U.S. They teach the overall concept, give a word problem, and then let the students work together to figure it out. Here is the interesting part, they leave the arithmetic for home or after school hours.
If a student wants (or parent) the child can stay after school to review or practice their arithmetic. They really don't believe in doing what we do in the U.S.... working constantly on arithmetic. They are very heavy into teaching the arithmetic, and then focusing heavily on a specific word problem format. I have studied both countries word problem fomats, both intracate but easy to learn. While they have many steps to problem solving, eventually, the students see it as only a few steps after they master the steps.
Oh, and on a later post.... I also have a workbook that combines literature with mathematics... I love the stories integrated with mathematics.
Justine
Arithmetic - a Local Perspective
By csonitaBob,
Right now in Westmoreland County, employers are begging our local Career Links to find trades industry employees who can still add, subtract, multiply, and divide in their head. Each one of these employers have their own specific scenarios for when this need occurs. However, since my husband is a cabinetmaker, I can share a specific cabinetmaking scenario:
A cabinetmaker receives a drawing from the designer of a doctor's office project. He is told that all the measurements are off because some one took inaccurate room measurements. There isn't time to re-draw (Why my husband says that "there is never enough time to do it right, but always enough time to do it over"). The cabinetmaker is provided with the accurate room dimensions and told to adjust all the drawing measurements accordingly as he builds the cabinets, desks, etc. Sometimes the difference is consistently off by the same fraction across the board - usually not. Either way, many employers expect the cabinetmaker to be able to make accurate adjustments mentally so that the installer does not have to make corrections on site (more costly). This is one of those situations in which the employer expects the worker to accurately and precisely calculate fractional or decimal changes mentally. By the way, that inaccurate measuring thing is why employers are also asking the local Career Links to find them employees who know how to use a measuring device accurately.
I'm concerned that providing math courses in which calculating is "invisible" could encourage some students to de-value the need to master basic arithmetic skills. I'm concerned because that is what I have seen occur on a local level. It is why I have learned to appreciate my son's Calculus teacher. Sometimes she lets them use the calculators; sometimes she doesn't. Sometimes she presents new concepts using Excel and a SmartBoard; sometimes she takes a more traditional approach. She facilitates mastery from every angle using an eclectic array of strategies.
Instead of separating concept mastery from calculation mastery, shouldn't we encourage students to appreciate how they work together to solve real world problems and assist with real world discoveries? Or am I really not understanding your thought process?
Chris
Math and Arithmetic
By anneBob,
My suggestions for teaching of math is to have students do more problem solving rather than just completing worksheets of numerical problems. And, the problems should be really "stories", with more facts and more numbers (perhaps in number words) than needed to answer the questions, which can be more than one.
I took a stab at creating such problems awhile back. They are called "Top of the Class" Word Problems in http://www.educationalsynthesis.org/math .... They begin at the beginning with making comparisons of numbers and I got as far as subtractions. Problems include former skills in the questions. I hope to put these into Javascript, but I have yet to develop the format for them. They will have variable names of characters, and perhaps variables for the things counted, and other stuff, as well as the random numbers for the solvable numbers.
Combining Spreadsheets & Word Problems
By csonitaI took a look at one of your word problem worksheets (comparing numerical values). I thought the worksheet had real potential. I especially liked having the student write their own word problem. If you do want to expand, I suggest you allow the teacher insert their local counties, grocery store names, etc. This would make the worksheets more relevant no matter where the student exists. I also think a collaboration betweeen Bob's spreadsheet idea and your word problem worksheets might be appreciated by those teachers trying to cover arithmetic and mathematics simultaneously.
Spreadsheets and word problems.
By drzIt is possible to customize the worksheets by having key words (name of student, name of city, name or store, etc) filled in by the student and then incorporate those words in the problem text through the use of range names. The value of range names in calculations is discussed in the Keyboarders Guide to Excel.
See: http://www.enabling.org/drupal/node/94
I see this as a two step process of learning the skill and applying the skill. In the learning section the numbers are kept relatively simple to teach the concept and in applying the skill the numbers can be more difficult but the arithmetic is invisible. It is the child's task to set up the problem in both cases, but in the application section, the arithmetic is handled by Excel.
Word Problems and spreadsheets
By anneBob & Chris,
It would be possibe to put the names in a range and randomly use them, or it would be possible to have the user/teacher/student, enter a set of names, kids, objects, stores, counties, etc., in javascript, and, along with random numbers create really sophisticated word problems that are relevent for Andrew to feel they pertain to him!
It just takes the time to build the word problems. I did a number which are linked as Top of the Class problems (my favorite is the one in which drz asks kids around the world about their brain preferences, and then the math student sums up drz's results a couple of different ways - I think it is on the addition word problems) but it takes time and creativity to think them up. I did a lot in a day of two and was drained of ideas. If I went back to them now, I should be able to come up with more. But, what would be best would be if all of us put our heads together and thought up scenarios in real life where comparing, adding, subtracting, multiplying and/or dividing of numbers answers intelligent questions. It would even be realistic to send the student out on a link, to, for example, see a chart with the dimensions of each planet
and then ask the student to make comparisons, sums, differences is those dimensions.
One of the thinks I developed as you look at the addition and subtraction problems, is that the problems had more than one question to answer. If two or three numbers were given, they may be asked which was the larger/largest, what the sum was, and what was the difference between the numbers, all from the same word problem.
This, Chris, is where I was getting into my idea for making a story that contained math problems in the story line and had to be solved "on the fly" while reading and comprehending the story.
Anne
Math Storybooks
By csonitaWhat a great idea! Expanding word problems into an entire story! After they have learned the concept and have begun to master the skill, they could apply that skill as they read their story. For younger students, they could apply what they have learned while learning about real life uses for math. For older students, the stories might also help with career exploration and/or life skills preparation. Teachers could use these stories as a type of pretest, enrichment, formative assessment, and/or summative assessment.
This is one of those times that I wish we lived closer. Bouncing story ideas across a table would really be useful.
Bob - What do you think?
Chris